There are many diverse encryption methods in the related art, and these methods have gained in commercial importance. They are used for transmitting information over generally accessible transmission media. However, only the owner of a cryptographic key is able to read this information in plain text.
A known method for establishing a common key via insecure communication channels is, for example, the W. Diffie and W. Hellman method (see DH method, W. Diffie and M. Hellmann; see “New Directions in Cryptography”, IEEE Transactions on Information Theory, IT-22(6):644-654, November 1976).
The Diffie-Hellmann key exchange [DH76] is based on the fact that it is virtually impossible to calculate logarithms modulo a large prime number p. Alice and Bob take advantage of this fact in the example illustrated below, by each secretly choosing a number x and y, respectively, smaller than p (and prime to p-1). They then send each other (consecutively or simultaneously) the x-th (and, respectively, y-th) power of a publicly known number α. From the received powers, they are able to calculate a common key K:=αxy by again performing an exponentiation with x and y, respectively. An attacker, who sees only αx and αy, is not able to calculate K therefrom. (The method known today to do so would involve first calculating the logarithm, e.g. of αx to the base a modulo p, and then raising αy to that power.)
